On the Transmittance of Metallic Superlattices in the Optical Regime and the True Refraction Angle

Pedro Pereyra

doi:10.3390/photonics8030086

Abstract

Transmission of electromagnetic fields through (dielectric/metallic)n superlattices, for frequencies below the plasma frequency ωp, is a subtle and important topic that is reviewed and further developed here. Recently, an approach for metallic superlattices based on the theory of finite periodic systems was published. Unlike most, if not all, of the published approaches that are valid in the n→∞ limit, the finite periodic systems approach is valid for any value of n, allows one to determine analytical expressions for scattering amplitudes and dispersion relations. It was shown that, for frequencies below ωp, large metallic-layer thickness, and electromagnetic fields moving along the so-called “true” angle, anomalous results with an apparent parity effect appear. We show here that these results are related to the lack of unitarity and the underlying phenomena of absorption and loss of energy. To solve this problem we present two compatible approaches, both based on the theory of finite periodic systems, which is not only more accurate, but has also the ability to reveal and predict the intra-subband resonances. In the first approach we show that by keeping complex angles, above and below ωp, the principle of flux conservation is fully satisfied. The results above ωp remain the same as in Pereyra (2020). This approach, free of assumptions, where all the information of the scattering process is preserved, gives us insight to improve the formalism where the assumption of electromagnetic fields moving along the real angles is made. In fact, we show that by taking into account the induced currents and the requirement of flux conservation, we end up with an improved approach, with new Fresnel and transmission coefficients, fully compatible with those of the complex-angle approach. The improved approach also allows one to evaluate the magnitude of the induced currents and the absorbed energy, as functions of the frequency and the superlattice parameters. We show that the resonant frequencies of intra-subband plasmons, which may be of interest for applications, in particular for biosensors, can be accurately determined. We also apply the approach for the transmission of electromagnetic wave packets, defined in the optical domain, and show that the predicted space-time positions agree extremely well with the actual positions of the wave packet centroids.

Highlights

We have shown the consequences that imposing the Bloch Theorem has on the transfer matrix method (TMM) and on the dispersion relations. This assumption is present in practically all the published works on infinite and semi-infinite metallic superlattices, even in quite recent publications, for example, Markos and Soukoulis’ book [76] and in the numerous papers published by Iakushev et al [77], all of them rely on the approach of Yeh et al We have shown that the most transcendental effects related to finiteness are, on one side, the possibility or not of defining quantities as the transmission and reflection coefficients and, on the other, very important from the point of view of plasmonic resonance applications, the ability or not to determine intra-subband frequency resonances
We have shown that the anomalous results and apparent parity effects reported in [1], are consequences of the common assumption that electromagnetic fields move along the direction of propagation of the constant-phase planes, the finiteness requirement, and the neglect of the induced currents
We fixed the proportionality constants by imposing the flux conservation requirement on the transfer matrices, which were built under the assumption that fields move along the real angle

Summary

Introduction

Note that many of the authors that impose Floquet’s theorem, which implies infinite systems, Photonics 2021, 8, 86 were forced to grapple with a kind of syncretic approach using Abelès finite periodic transfer matrix, in order to calculate the transmission coefficients. Unlike these approaches, the theory of finite periodic system (TFPS) was able to determine the dispersion relation keeping the number of unit cells as an essential condition. A detailed analysis of the space-time evolution of Gaussian wave packets will be publish elsewhere

Unitarity Deficit in the Constant-Phase Direction

Criticism and Differences with Infinite Superlattices Approaches

Complex-Angle Approach

Improved Real-Angle Approach

Reflection and Transmission of Gaussian Pulses by Metallic Superlattices

Conclusions